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I am working on an academic physics problem, where I am trying to determine favourable variables in two data sets, where both data sets have the same variables.

The most favourable variables would have the highest difference in means and a small dispersion of data - a small 'spread'.

Is there a statistical measure/metric that combines both the difference in means and dispersion of data in this way?

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  • $\begingroup$ A standardised effect size like Cohen's d is probably what you want. en.wikiversity.org/wiki/Cohen%27s_d $\endgroup$
    – George Savva
    Mar 5, 2020 at 12:35
  • $\begingroup$ This page on standardised effects has more discussion. en.wikipedia.org/wiki/… $\endgroup$
    – George Savva
    Mar 5, 2020 at 12:42
  • $\begingroup$ And t-statistic (from two sample t.test) may fit your goal. $\endgroup$
    – Pere
    Mar 5, 2020 at 12:57
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    $\begingroup$ This is a valuation problem: one in which you must make a trade-off between an increased difference in means and a decrease in dispersion. Accepting some solution that has been used in other applications would amount to supposing your values are comparable to those of the other applications--which, even if plausible, ought to be checked. The most general solution will be of the form $\alpha f(m)-\beta g(d)$ where $f$ and $g$ are non-decreasing functions, $\alpha$ and $\beta$ are non-negative numbers, and $m$ is the mean difference and $d$ is some measurement of the dispersion. $\endgroup$
    – whuber ♦
    Mar 5, 2020 at 16:13
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    $\begingroup$ (Continued) In light of this, what information can you add to your question that will help us understand, quantitatively, how you value these potential trade-offs? $\endgroup$
    – whuber ♦
    Mar 5, 2020 at 16:14

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Standardised effect sizes scale the difference between groups by the variance within each group. Cohen's $d$ is probably the simplest and most widely used. Cohen's $d$ is also sometimes called the 'standardised mean difference'.

To calculate it simply divide the difference in means between the groups by the pooled standard deviation estimated within the groups.

Note this is different to the t-statistic which is the difference in means divided by the standard error of the difference in means. This will depend heavily on sample size (bigger sample sizes will lead to higher t-statistics), while Cohen's d will not depend on sample size (except that it becomes more precisely estimated as sample size increases).

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  • $\begingroup$ Thanks George. This is exactly what I am after. $\endgroup$
    – Cairan Van Rooyen
    Mar 6, 2020 at 12:56
  • $\begingroup$ No problem. @whuber makes an interesting point in the comments though. $\endgroup$
    – George Savva
    Mar 6, 2020 at 13:14

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